Boundedly Compact Space
A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the Bolzano-Weierstrass theorem.)
Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.
See also
Bolzano-Weierstrass Theorem, Bounded, Complete Metric Space, Hopf-Rinow Theorem, Metric SpaceExplore with Wolfram|Alpha
WolframAlpha
More things to try:
Cite this as:
Weisstein, Eric W. "Boundedly Compact Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BoundedlyCompactSpace.html